Odd Spheres Research Articles

Rigidity on quantum symmetry for a certain class of graph C*-algebras

Quantum symmetry of graph C*-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group (C(S1)∗C(S1)∗⋯∗C(S1)︸|E(Γ)|−times,Δ) [in short, *|E(Γ)|C(S1),Δ] always acts on a graph C*-algebra for a finite, connected, directed graph Γ in the category introduced by Joardar and Mandal, where |E(Γ)| ≔ number of edges in Γ. In this article, we show that for a certain class of graphs including Toeplitz algebra, quantum odd sphere, matrix algebra etc. the quantum symmetry of their associated graph C*-algebras remains *|E(Γ)|C(S1),Δ in the category as mentioned before. More precisely, if a finite, connected, directed graph Γ satisfies the following graph theoretic properties: (i) there does not exist any cycle of length ≥2 (ii) there exists a path of length (|V(Γ)| − 1) which consists all the vertices, where |V(Γ)| ≔ number of vertices in Γ (iii) given any two vertices (may not be distinct) there exists at most one edge joining them, then the universal object coincides with *|E(Γ)|C(S1),Δ. Furthermore, we have pointed out a few counter examples whenever the above assumptions are violated.

Sequences of operator algebras converging to odd spheres in the quantum Gromov–Hausdorff distance

Sequences of operator algebras converging to odd spheres in the quantum Gromov–Hausdorff distance

Homotopy Theoretic Properties Of Open Books

ABSTRACT We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions we prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor’s open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.

Relative Gottlieb groups of mapping spaces and their rational cohomology

Let f:X →Y be a map of simply connected CW-complexes of finite type. Put maxπ★(Y)⊗Q = max{ i | πi(Y)⊗Q≠0 }. In this paper we compute the relative Gottlieb groups of f when X is an F0-space and Y is a product of odd spheres. Also, under reasonable hypothesis, we determine these groups when X is a product of odd spheres and Y is an F0-space. As a consequence, we show that the rationalized G-sequence associated to f splits into a short exact sequence. Finally, we prove that the rational cohomology of map(X,Y;f) is infinite dimensional whenever maxπ★(Y)⊗Q is even.

Hopf coactions on odd spheres

We prove that the q-deformed unitary group, i.e., Uq(N), is the universal compact quantum group in the category of (compact) quantum groups which coact on the q-deformed odd sphere Sq2N−1 leaving the space spanned by the natural set of generators invariant and preserving the unique SUq(N) invariant functional on Sq2N−1. Using this, we identify Uq(N) as the quantum group of orientation preserving isometries (in the sense of Bhowmick and Goswami [5]) for a natural spectral triple associated with Sq2N−1 constructed by Chakraborty and Pal [9].

Symmetrized topological complexity

We present upper and lower bounds for symmetrized topological complexity [Formula: see text] in the sense of Basabe–González–Rudyak–Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of the symmetric square [Formula: see text]. We also show that symmetrized topological complexity coincides with its monoidal version, where the path from a point to itself is required to be constant. Using these results, we calculate the symmetrized topological complexity of all odd spheres.

Rational homotopy of mapping spaces between complex Grassmannians

The complex Grassmann Gr(k, n) is the space of k dimensional subspaces of ℂ n . It is a complex manifold of complex dimension k(n − k). There is a natural inclusion ik ,n : Gr(k, n) ↪ Gr(k, n + r). In this paper, we use Sullivan models to compute the rational homotopy type of the component of the inclusion Gr(2, n) ↪ Gr(2, n + r) in the space of mappings from Gr(2, n) to Gr(2, n + r), r ≥ 1. We show in particular that map(Gr(2, n), Gr(2, n + 1); in ) has the rational homotopy type of a product of odd spheres.

The $K$-theory of twisted multipullback quantum odd spheres and complex projective spaces

We find multipullback quantum odd-dimensional spheres equipped with natural U(1) -actions that yield the multipullback quantum complex projective spaces constructed from Toeplitz cubes as noncommutative quotients. We prove that the noncommutative line bundles associated to multipullback quantum odd spheres are pairwise stably non -isomorphic, and that the K -groups of multipullback quantum complex projective spaces and odd spheres coincide with their classical counterparts. We show that these K -groups remain the same for more general twisted versions of our quantum odd spheres and complex projective spaces.

Odd sphere bundles, symplectic manifolds, and their intersection theory

Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of Aoo-algebras built of differential forms on the symplectic manifold. We show that these symplectic Aoo-algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these Aoo-algebras are equivalent to the standard de Rham differential graded algebra on certain odd-dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yau's symplectic Aoo-algebras satisfy the Calabi-Yau property, and importantly, that they can be used to define an intersection theory for coisotropic/isotropic chains. We further demonstrate that these symplectic Aoo-algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality and invariance in the smooth category.

Smooth crossed product of minimal unique ergodic diffeomorphism of odd sphere

For minimal unique ergodic diffeomorphisms \alpha_n of S^{2n+1} (n > 0) and \alpha_m of S^{2m+1}(m>0) , the C^* -crossed product algebra C(S^{2n+1})\rtimes_{\alpha_n} \mathbb{Z} is isomorphic to C(S^{2m+1})\rtimes_{\alpha_m} \mathbb{Z} even though n\neq m . However, by cyclic cohomology, we show that smooth crossed product algebra C^\infty(S^{2n+1})\rtimes_{\alpha_n} \mathbb{Z} is not isomorphic to C^\infty(S^{2m+1})\rtimes_{\alpha_m} \mathbb{Z} if n\neq m .

Free energy on a cycle graph and trigonometric deformation of heat kernel traces on odd spheres

We consider a possible ‘deformation’ of the trace of the heat kernel on odd dimensional spheres, motivated by the calculation of the free energy of a scalar field on a discretized circle. By using an expansion in terms of the modified Bessel functions, we obtain the values of the free energies after a suitable regularization.

Quantum Hall effect on odd spheres

We solve the Landau problem for charged particles on odd-dimensional spheres $S^{2k-1}$ in the background of constant SO(2k-1) gauge fields carrying the irreducible representation $\left ( \frac{I}{2}, \frac{I}{2}, \cdots, \frac{I}{2} \right)$. We determine the spectrum of the Hamiltonian, the degeneracy of the Landau levels and give the eigenstates in terms of the Wigner ${\cal D}$-functions, and for odd values of $I$ the explicit local form of the wave functions in the lowest Landau level (LLL). Spectrum of the Dirac operator on $S^{2k-1}$ in the same gauge field background together with its degeneracies is also determined and in particular the number of zero modes is found. We show how the essential differential geometric structure of the Landau problem on the equatorial $S^{2k-2}$ is captured by constructing the relevant projective modules. For the Landau problem on $S^5$, we demonstrate an exact correspondence between the union of Hilbert spaces of LLL's with $I$ ranging from $0$ to $I_{max} = 2K$ or $I_{max} = 2K+1$ to the Hilbert spaces of the fuzzy ${\mathbb C}P^3$ or that of winding number $\pm1$ line bundles over ${\mathbb C}P^3$ at level $K$, respectively.

Quantum revivals in free field CFT

The recent work by Cardy (arXiv:1603.08267) on quantum revivals and higher dimensional CFT is revisited and enlarged upon for free fields. The expressions for the free energy used here are those derived some time ago. The calculation is extended to spin–half fields for which the power spectrum involves the odd divisor function. An explanation of the rational revivals for odd spheres is given in terms of wrongly quantised fields and modular transformations. Comments are made on the equivalence of operator counting and eigenvalue methods, which is quickly verified.

On the free loop spaces of a toric space

In this note, it is shown that the Hilbert–Poincare series for the rational homology of the free loop space on a moment-angle complex is a rational function if and only if the moment-angle complex is a product of odd spheres and a disk. A partial result is included for the Davis–Januszkiewicz spaces. The opportunity is taken to correct the result (Bahri et al., Proceedings of the Steklov Institute of Mathematics, Russian Academy of Sciences, vol. 286, pp. 219–223. doi:10.1134/S0081543814060121, 2014) which used a theorem from Berglund and Jollenbeck (J Algebra 315:249–273, 2007).

Mod $p$ Decomposition of $H$-spaces of Low Rank

Let X be a mod p H -space such that the mod p cohomology is an exterior algebra generated by finitely many generators of degree (2n_1+1, 2n_2+1, \dots, 2n_k+1) with 1 \le n_1 \le n_2 \le \cdots \le n_k . It is known that if n_k-n_1 < p-1 then X decomposes to a product of odd spheres, and if n_k-n_1 < 2 (p-1) then X decomposes to a product of odd spheres and B_n(p) s. In the paper we consider the case of n_k-n_1 < 3 (p-1) , and give a product decomposition of X to irreducible factors.

Constructing the quantum Hall system on the Grassmannians Gr2(CN)

In this talk, we give the formulation of Quantum Hall Effects (QHEs) on the complex Grassmann manifolds Gr2(CN). We set up the Landau problem in Gr2(CN), solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the SU(N) Wigner D-functions for charged particles on Gr2(CN) under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. For the simplest case of Gr2(C4) we provide explicit constructions of the single and many- particle wavefunctions by introducing the Plucker coordinates and show by calculating the two-point correlation function that the lowest Landau level (LLL) at filling factor v = 1 forms an incompressible fluid. Finally, we heuristically identify a relation between the U(1) Hall effect on Gr2(C4) and the Hall effect on the odd sphere S5, which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on CP3 and CP7 and those on the spheres S4 and S8, respectively. The talk is given by S. Kürkçüoğlu at the Group 30 meeting at Ghent University, Ghent, Belgium in July 2014 and based on the article by F.Ballı, A.Behtash, S. Kürkçüoğlu, G.Ünal [1].

On the rational type of moment-angle complexes

In this note it is shown that the moment angle complexes Z(K;(D^2,,S^1)) which are rationally elliptic are a product of odd spheres and a disk

Cellular decomposition of quaternionic spherical space forms

We obtain an explicit cellular decomposition of the quaternionic spherical space forms, manifolds of positive constant curvature that are factors of an odd sphere by a free orthogonal action of a generalized quaternionic group. The cellular structure gives and explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group. As an application we compute the Whitehead torsion of these spaces for any representation of the fundamental group.

Examples of Rational Toral Rank Complex

There is a CW complex 𝒯(𝑋), which gives a rational homotopical classification of almost free toral actions on spaces in the rational homotopy type of X associated with rational toral ranks and also presents certain relations in them. We call it the rational toral rank complex of X. It represents a variety of toral actions. In this note, we will give effective 2-dimensional examples of it when X is a finite product of odd spheres. This is a combinatorial approach in rational homotopy theory.

Spherical Casimir pistons

A piston is introduced into a spherical lune Casimir cavity turning it into two adjacent lunes separated by the (hemispherical) piston. On the basis of zeta-function regularization, the vacuum energy of the arrangement is finite for conformal propagation in spacetime. For even spheres this energy is independent of the angle of the lune. For odd dimensions it is shown that for all Neumann, or all Dirichlet, boundary conditions the piston is repelled or attracted by the nearest wall if d = 3, 7, … or if d = 1, 5, … , respectively. For hybrid N–D conditions these requirements are switched. If a mass is added, divergences arise which render the model suspect. The analysis, however, is relatively straightforward and involves the Barnes zeta function. The extension to finite temperatures is made and it is shown that for the 3, 7, … series of odd spheres, the repulsion by the walls continues but that, above a certain temperature, the free energy acquires two minima symmetrically placed about the midpoint.